Strong divergence of approximation processes in banach spaces

In this paper we analyze the divergence behavior of linear approximation processes in general Banach spaces. If the approximation process is given by a sequence of bounded linear operators, often the Banach–Steinhaus theorem is employed to perform a divergence analysis. However, the Banach– Steinhaus theorem gives only divergence in terms of the limit superior, and no strong divergence in terms of the limit. Strong divergence is closely related to the question of whether adaptive approaches, where the approximation process is adapted to the signal, can be successful. Here we study the structure of the set of signals for which the approximation process is strongly divergent, i.e., does not have a convergent subsequence, and show that, under mild conditions, this set is at most a meager set. Further, for the Shannon sampling series, which is a special case of the general theory, we prove that the set of signals in PW¹_π with strong divergence is lineable. That is, there exists an infinite dimensional subspace such that we have strong divergence for all signals, except the zero signal, from this subspace.